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Interesting Factorial Sum

I wanted to share this sum with you all, though I have no clue how to prove it ---

\[\sum_{n=1}^{\infty}\left[(-1)^n \dfrac{(n!)^2}{n^2(2n)!}\right]=2\log(\phi)\log\left(\dfrac{1}{\phi}\right)\] where \(\phi\) is the golden ratio.

It has an amazing closed form, doesn't it?

Note by Pratik Shastri
3 years ago

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http://mathworld.wolfram.com/CentralBinomialCoefficient.HTML gives a bit of info

Bogdan Simeonov - 3 years ago

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Here's a proof for a lot of these( including this one ): click me

Bogdan Simeonov - 3 years ago

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